Liquid Air Cycle Rocket Equation

Henry Spencer Comment

To simplify
our derivation we assume that the rocket is frictionless (except for engine
drag)...

Fairly unrealistic for an airbreather, alas.

The June 1994 JBIS had a very interesting paper by Bob Zubrin, suggesting
a methane-fueled variant of NASP. (The trick was that he used friction heat
to turn the methane into an acetylene/hydrogen mix before burning it.) What
I found more interesting, though, was his derivation of an extended version
of the rocket equation for high-speed airbreathers.

The one key assumption required is that you have to know how the specific
impulse varies with speed. Turns out, though, that a relatively good
approximation -- valid from turbojets to scramjets, starting at about Mach
2 and going up a long way from there -- is to simply assume that Isp is
inversely proportional to speed.

Oh yes, we also assume a standing start, V_initial = 0.

Given that, with some algebraic manipulation one ends up with a fairly
simple equation. I'll state it a bit differently than he did:

V_final = Isp_half * g * ln(mass_ratio) * 1/(1 + 1/(L/D * A/g))

Isp_half is the Isp at V_final/2. This in itself is interesting -- it
says that the huge Isp advantages to be had at low speeds buy you very
little, because so much of your accelerating is done at high speeds
where airbreathing Isp falls off badly, to 1000s or so.

The really fun part, though, is that last term. The second term in its
denominator, 1/(L/D * A/g), is what you might call the Air Breather's
Burden. L/D is average lift/drag, a familiar basic measure of aerodynamic
performance for winged vehicles. And A/g is just average forward
acceleration in Gs. If either of these goes to infinity -- you have
either miraculous aerodynamics or tremendous acceleration -- the value of
the ABB goes to 0, the value of the whole last term goes to 1, and you get
the familiar rocket equation.

But if they don't go to infinity, what you get is trouble. Hypersonic
L/D ratios typically are not good. Zubrin guesses L/D of 5 for NASP
and 7 for his design (better because methane tanks are more compact
than hydrogen tanks, permitting a slimmer, less draggy shape). If
NASP's average acceleration is 0.2G -- nothing for rockets, but fairly
impressive for airbreathers at high speeds -- the ABB equals 1.0, and
NASP gets only about half the V_final you would predict from the bare
rocket equation. This reduces its effective Isp to about 500s, which
means it needs nearly the same mass ratio as a good oxyhydrogen rocket
SSTO, despite not having to carry any oxidizer. (Worse, that means it
needs *seven times* as much hydrogen, since the LOX/LH2 ratio is 6:1
for the rocket.)

Even if we assume an acceleration of 0.5G as Zubrin suggests -- which is
really fierce acceleration for an airbreather, given how lousy the T:W
ratio of airbreathing engines is -- the ABB only drops to 0.4 and the
airbreather still needs 40% more delta-V than the rocket.

There is more to life than getting a job | Henry Spencer
and making a living. --Barbara Morgan | henry@zoo.toronto.edu